#8993480

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4525 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

(1,2)(3,6)(4,5)(7,8)(9,10,14,15)(11,17,19,13)(12,20,21,16)(18,23,24,22)(25,26)(27,32)(28,31)(29,33)(30,34)(35,37)(36,39)(38,40);(1,3,5,7)(2,6,4,8)(9,11,14,19)(10,13,15,17)(12,18,21,24)(16,23,20,22)(25,27)(26,28)(29,35)(30,31)(32,34)(33,36)(37,40)(38,39);(2,4)(3,7)(10,15)(12,21)(13,17)(18,24)(25,28,30,32)(26,31,34,27)(29,37,38,36)(33,35,40,39);(1,3,5,7)(2,6,4,8)(9,12)(10,16)(11,18)(13,23)(14,21)(15,20)(17,22)(19,24)(25,28)(26,27)(29,36)(30,32)(31,34)(33,35)(37,38)(39,40);(1,3)(2,6)(4,8)(5,7)(9,13)(10,11)(12,22)(14,17)(15,19)(16,24)(18,20)(21,23)(25,27)(26,32)(28,34)(29,35)(30,31)(33,37)(36,40)(38,39);(1,4,5,2)(3,8,7,6)(9,10)(11,13)(12,16)(14,15)(17,19)(18,23)(20,21)(22,24)(25,29,30,38)(26,33,34,40)(27,35,31,39)(28,36,32,37)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=ghi,d^2=i,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8900676);